5.7.1 Nelson-Siegel-Svensson model

The Nelson-Siegel-Svensson (NSS) model (Medvedev 2019) is an extended version of the Nelson-Siegel (NS) model (Annaert et al. 2013), commonly used to describe interest rate term structures or yield curves. A yield curve represents the relationship between the interest rate (or cost of borrowing) and the time to maturity of the debt for a given borrower in each currency.

The NSS model defines the instantaneous forward rate as a function of four components: a long-term factor, a short-term factor, a medium-term factor, and an over-demand factor:

$$Y(t)= β_0 + \frac{β_1 (1-e^{-t/τ_1})}{t/τ_1 } + β_2\left( \frac{1-e^{-t/τ_1}}{t/τ_1} -e^{-t/τ_1} \right) + β_3 \left( \frac{1-e^{-t/τ_2}}{t/τ_2} - e^{-t/τ_2} \right) = Y_0+\\+Y_1+Y_2+Y_3 \ \ \ \ \ \ \ \ \tag{40}$$

Here's what each component represents:

o (Long term factor): This component, represented by , is the long-term interest rate or the level factor of the yield curve. It is a constant value that sets the asymptotic future interest rate as time tends to infinity.

o (Short term factor): This component captures the impact of short-term rates and is responsible for the initial steepness of the yield curve. The parameter sets the slope at time zero and represents a decay factor. The term introduces a decay effect, meaning the influence of this factor decreases as time progresses.

o (Medium term factor): This component models the hump in the yield curve. sets the size of the hump, while again is a decay factor. The term means the influence of this factor increases initially, reaches a peak, and then decreases.

o (Over-demand factor): This component, unique to the NSS model, captures any additional influences on the interest rate not covered by the other three components. sets the size of this influence, while acts as a decay factor. Like , the term introduces an increase, peak, and decrease effect.

The final yield curve is the sum of these four components, yielding a flexible model that can capture a variety of yield curve shapes.

Figure 91 Adjusted NSS Model over time